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--> [Submitted on 27 Apr 2026] Title:On Hamming-Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs View a PDF of the paper titled On Hamming-Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs, by Alokendu Mazumder and 2 other authors View PDF HTML (experimental) Abstract:The subdominant (minmax) ultrametric is a canonical tree-structured summary of a dissimilarity matrix, arising equivalently as the ultrametric induced by single-linkage clustering. While its classical stability theory is usually formulated in $\ell_\infty$ or Gromov--Hausdorff terms, such bounds are poorly suited to sparse perturbations that alter only a few pairwise distances. We develop an $\ell_0$-type stability theory for this operator. Our analysis shows that sparse edits propagate only through the minimum spanning tree (MST): a pairwise ultrametric value can change only if its tree path crosses an edited edge or a cut newly exposed by an edited off-tree edge. This yields a sharp per-edit exposed-cut score and a tree-only global envelope, leading to Hamming--Lipschitz bounds on the number of ultrametric entries that can change. We also prove sharpness results showing that this dependence on tree geometry is unavoidable: under strict cut separation the tree-edge bound is attained exactly, and for off-tree edits there are explicit families in which one edited distance changes $\Theta(n^2)$ ultrametric entries. In addition, we prove a conditional near-additivity principle for multiple edits under certified large per-edit changed regions and negligible aggregate overlap. Experiments on deep-embedding graphs show that the resulting structural scores provide useful vulnerability diagnostics for hierarchical representations. Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI) Cite as: arXiv:2608.04014 [cs.LG] (or arXiv:2608.04014v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2608.04014 arXiv-issued DOI via DataCite Submission history From: Alokendu Mazumder [view email] [v1] Mon, 27 Apr 2026 12:16:34 UTC (5,543 KB) Full-text links: Access Paper: View a PDF of the paper titled On Hamming-Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs, by Alokendu Mazumder and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-08 Change to browse by: cs cs.AI References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)